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Machine-learned potentials link vacancy-dependent copper transport to post-implantation profiles in silicon

This study employs a machine-learned deep potential framework to demonstrate how vacancy-dependent copper migration governs post-implantation redistribution in silicon, offering a significantly more accurate predictive model for annealed depth profiles than traditional methods.

Original authors: Mingzhu Liu, Yiwen Su, Xinzhong Wang, Jiashu Chen

Published 2026-10-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mingzhu Liu, Yiwen Su, Xinzhong Wang, Jiashu Chen

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Silicon is the silent foundation of the modern world, the crystalline material that powers everything from smartphones to solar panels. For these devices to work, the silicon must be pure, yet it is constantly under threat from microscopic invaders. Copper, a metal prized for its ability to carry electricity efficiently in the wires connecting computer chips, is also a dangerous contaminant when it slips into the silicon itself. Even a tiny amount of copper can wander through the crystal lattice, creating electrical shortcuts that cause devices to fail or leak energy. The problem is that copper moves with surprising speed, especially when heated during the manufacturing process, and it does not travel alone. It interacts with empty spaces in the crystal structure, known as vacancies, which act like traps or hiding spots. Understanding exactly how copper moves, gets stuck, and spreads through silicon is critical for engineers trying to build faster, more reliable electronics, but the process happens at a scale so small and moves so quickly that it is nearly impossible to watch directly with a microscope.

To solve this puzzle, a team of researchers from Central South University and the Hong Kong Polytechnic University turned to a powerful combination of computer simulations. They did not rely on old, simplified models that guess how atoms behave. Instead, they built a new, highly accurate digital model of the silicon and copper atoms using a method called machine learning. This approach allowed them to train a computer program to learn the rules of atomic interaction directly from the most precise physics calculations available, creating a "deep potential" that could predict how atoms move with near-perfect accuracy. With this tool in hand, they simulated the entire journey of copper: first, how it is forced into the silicon during manufacturing, and second, how it spreads out when the material is heated to repair the damage.

The researchers started by simulating the moment copper ions are shot into the silicon, a process called ion implantation. In the real world, this creates a messy, uneven distribution of copper atoms deep inside the material. The team's new simulation captured this initial state with remarkable detail, matching real-world measurements far better than previous methods. They then let the simulated silicon sit in a virtual oven, heating it up to see how the copper would redistribute. In the absence of any empty spaces in the crystal, the copper atoms zipped around freely, moving from one spot to another with very little resistance. The simulation showed that these atoms could travel between specific positions in the crystal lattice with an energy barrier of just 0.185 electronvolts, a value that matched the most trusted theoretical predictions. This confirmed that the digital model was accurate enough to trust for the next, more complex step.

The real discovery came when the researchers introduced empty spaces, or vacancies, into their silicon crystal. In the real world, the process of implanting copper often creates these vacancies, but their exact number and location are hard to pin down. The team created four different versions of their simulation, each containing a different number of these empty spots, ranging from none to eight. They watched what happened to the eight copper atoms in each scenario. The results were striking. In the crystal with no vacancies, the copper moved quickly and evenly. But as soon as vacancies were introduced, the behavior changed completely. The copper atoms began to get caught. Some would zip across the crystal for a while, only to dive into an empty spot and stay there, effectively becoming stuck. The more vacancies present, the more copper atoms became trapped, and the slower the overall group moved.

This trapping effect created a complex, mixed behavior that simple models had missed. In the simulations with many vacancies, the copper atoms did not move as a uniform group. Instead, some were still racing around while others were sitting still, waiting to be released. This meant that the average speed of the copper dropped significantly, and the way it spread out became irregular and unpredictable. The researchers found that the presence of these traps changed the apparent energy required for the copper to move, making it look like the copper was moving more easily in some ways, even though it was actually being held back. The simulation revealed that the copper was constantly switching between being a fast, free-moving traveler and a stationary prisoner of the crystal defects.

To see if this microscopic behavior matched reality, the team compared their final results with actual measurements taken from real silicon samples. They looked at how deep the copper had penetrated and how it had spread out after heating. The new machine-learning approach predicted the shape of the copper distribution with much higher accuracy than the standard methods used in the industry. It correctly described not just the main cluster of copper, but also the faint, thin trail of copper atoms that had wandered far away from the main group. This tail of contamination is particularly dangerous because it can reach sensitive parts of a chip that the main cluster does not touch. The new method reduced the error in predicting these profiles by nearly half compared to older techniques, proving that accounting for the specific dance between copper and vacancies is essential for getting the picture right.

The study concludes that to control copper contamination, engineers cannot just look at the temperature or the initial amount of copper. They must also understand the hidden landscape of empty spaces inside the silicon. The copper does not just diffuse; it gets trapped, released, and re-trapped in a complex cycle that depends entirely on the defects created during manufacturing. By linking the tiny, atomic-scale interactions to the large-scale patterns seen in real devices, this new framework offers a clearer path to predicting and preventing copper damage. It suggests that by carefully managing the creation of these vacancies during the implantation process, manufacturers might be able to keep the copper in check, ensuring that the silicon remains pure enough to power the next generation of technology.

Technical Summary: Machine-learned potentials link vacancy-dependent copper transport to post-implantation profiles in silicon

Problem Statement
Copper (Cu) contamination poses a critical reliability threat to silicon (Si) devices, particularly in advanced integrated circuits and through-silicon vias. Mobile interstitial Cu can rapidly diffuse through the Si lattice, forming electrically active defects, precipitates, and silicide phases that degrade junction performance and increase leakage currents. While the intrinsic interstitial diffusion of Cu is known to be fast (migration barrier ~0.18 eV), macroscopic transport is complicated by interactions with lattice defects, specifically Si vacancies. These vacancies act as kinetic traps, converting mobile interstitial Cu into immobile substitutional configurations or complexes. Current experimental methods (e.g., SIMS) provide spatially averaged profiles but cannot resolve individual capture and release events. Conversely, traditional atomistic simulations face a trade-off: Density Functional Theory (DFT) offers accuracy but is limited to small systems and short timescales, while classical molecular dynamics (MD) lacks the transferability to accurately describe complex Cu-defect energetics. There is a need for a framework that connects elementary Cu migration mechanisms with vacancy-dependent transport to accurately predict post-implantation depth profiles.

Methodology
The authors developed a multiscale framework integrating DFT-informed implantation simulations with an actively learned Deep Potential (DP) model for large-scale molecular dynamics.

Deep Potential Construction: An ensemble of deep neural network potentials was trained using the DP-GEN concurrent-learning scheme. The training dataset included interstitial and substitutional Cu configurations, vacancy-containing structures, and migration pathways, augmented by cell deformations and random perturbations. The model was iteratively refined by identifying uncertain configurations for additional DFT labeling until model uncertainty converged.
Validation: The DP model's accuracy was rigorously tested against independent DFT datasets for Si-Cu systems (Si64Cu, Si63Cu, Si62Cu2). Additionally, Nudged-Elastic-Band (NEB) calculations were performed using both DFT and the trained DP to compare migration barriers and vacancy capture energetics.
Implantation and Annealing Simulations:
- Implantation: DFT-MD was used to simulate 70 keV Cu ion implantation into Si, generating atomistically resolved initial configurations without assuming predefined dopant distributions.
- Diffusion: The trained DP model enabled large-scale DeePMD simulations (4,096 Si atoms + 8 Cu atoms) over 600–1400 K. Four distinct systems were modeled containing 0, 2, 4, or 8 Si vacancies to isolate the effect of vacancy concentration on transport.
- Analysis: Cu depth profiles were generated before and after thermal annealing and compared against Secondary-Ion Mass Spectrometry (SIMS) experimental data and a conventional Monte Carlo/Binary Collision Approximation + Kinetic Monte Carlo (MC/BCA+KMC) workflow.

Key Results

Model Accuracy: The DP model reproduced DFT energies with Root-Mean-Square Errors (RMSE) of 3.16–3.71 meV/atom and forces with RMSEs of 35.10–88.30 meV/Å across various Cu-Si environments.
Migration and Trapping Energetics:
- The DP-calculated migration barrier for interstitial Cu between tetrahedral sites is 0.192 eV, closely matching the DFT value of 0.185 eV.
- Vacancy capture is highly exothermic; the transition from interstitial Cu to substitutional Cu (Cui + VSi → Cus) releases approximately 2.5 eV, confirming vacancies as strong energetic sinks.
- The model accurately reproduces the energetics of Cu-vacancy complex formation (e.g., ⟨Cus, Cui⟩ + VSi → ⟨Cus, Cus⟩).
Vacancy-Dependent Transport:
- In vacancy-free Si, the calculated activation energy is 0.202 eV, consistent with experimental intrinsic diffusion values.
- As vacancy concentration increases (V2, V4, V8), the ensemble-averaged diffusivity decreases, and the apparent activation energy drops (to 0.168, 0.087, and 0.083 eV, respectively).
- Atom-resolved trajectories reveal a competition between rapid interstitial migration and prolonged residence in vacancy-associated states. The decline in Arrhenius fit quality ( $R^2$ ) for vacancy-rich systems indicates that a single activation barrier cannot describe the heterogeneous population of mobile and trapped Cu atoms.
Profile Prediction:
- The DFT-MD+DP framework significantly outperforms the MC/BCA+KMC workflow in predicting Cu depth profiles.
- For the full profile (0–900 Å), the DP method reduced the mean-normalized Normalized Root-Mean-Square Error (NRMSE) by 47–61% across various annealing temperatures compared to MC/BCA+KMC.
- Improvements were observed in both the near-surface region (0–200 Å) and the low-concentration diffusion tail (700–900 Å). The DP profiles retained approximately 90% of the experimental tail inventory, whereas the MC/BCA+KMC profiles retained only 73–79%.

Significance and Claims
The paper claims to provide a first-principles-informed framework that successfully links elementary Cu-defect interactions to measurable macroscopic redistribution. By combining DFT accuracy with the scalability of machine-learned potentials, the authors demonstrate that:

Mechanistic Insight: Vacancy populations act as critical controls on Cu redistribution, suppressing the mobile Cu fraction and creating spatially heterogeneous dynamics that cannot be captured by a single diffusion coefficient.
Predictive Capability: The framework offers a route to jointly evaluate implantation-induced defects and annealing conditions. It provides a more accurate prediction of Cu contamination profiles than traditional MC/BCA+KMC methods, particularly in the low-concentration tails where defect trapping and release are critical.
Methodological Advancement: The study validates that active-learning DP models can bridge the gap between small-scale DFT energetics and large-scale process-relevant simulations, enabling the study of defect-mediated diffusion in complex, non-equilibrium environments without prescribing a fixed catalogue of migration events.

The authors maintain a modest stance, noting that while the framework improves profile prediction, longer simulations are required to fully resolve the kinetics of repeated capture and release events. They position their work as a complementary capability to existing KMC models, specifically addressing the challenge of evolving defect configurations and correlated Cu-vacancy interactions in silicon processing.

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